I'm confused. This 'button' I a website feature. As you can read, on the website you linked, you type 'your' own skype name there, it generates a HTML code, and you copy and paste that code to your own website. So then anyone that visites your website can easily send you a message, call or whatever.
I do not understand what is the purpose of creating a launcher on your desktop with 'your friends' skype name. You already have it in skype application.
This button is for other people to contact you and not the other way.